jane turner [4246] phy 316 (2003 spring) lecture 3 homework/quiz etc problematic email addresses:...

Jane Turner [4246] PHY 316 (2003 Spring) Lecture 3

Homework/Quiz etc

Problematic email addresses:[email protected] [email protected] [email protected] (this one ok, I have an alternative) -Mailbox full errors for these

A class email noted L1 + L2 and answer to homework on the webpage http://www.jca.umbc.edu/~turner/2003_phys316.html

Homework will not be graded, but same and similar questions will appear in exams and quizzes.

QUIZ next Tuesday -at start of lecture, 45 minutes. Bring a calculator. Will cover last weeks, todays and some of Thursdays material. Closed book!


Review of Kepler

Kepler’s First Law: The orbit of each planet about the Sun is an ellipse with the Sun at one focus


Orbits


Review of Kepler

Kepler’s Second Law: As a planet moves in its orbit around the Sun, it sweeps out equal areas in equal times


Review of Kepler

Kepler’s Third Law: P2=R3 where P is the period in years & R is the semi-major axis in AU (1 AU is Earth-Sun dist)


Review of Kepler

So, Kepler parameterized the planetary orbits, without understanding why things were as they are…

and P2=R3 only true using certain units (years and AU, all Earth-related measures)

-in general P2= k R3 and the physics hidden in the constant, k, was not understood.


Galileo Galilei - The Telescope

At the same time as Kepler proposed his laws of planetary motion, Galileo was proving planets really did move around the Sun, supporting Copernicus model

Circa 1600, - still 3 objections to a heliocentric solar system with noncircular orbits. All were rooted in the beliefs of Aristotle.

-The Earth could not be moving because if it were, all the objects on it would be left behind as it moved.

-The heavens were thought to be perfect and unchanging

-If the Earth orbits the sun, we ought to be able to see stellar parallax.



In 1609 Galileo Galilei starts using the telescope for astronomy-took the basic telescope used for terrestrial viewing and turned it into a scientific instrument



Letter from Galileoreporting the discoveryof Jupiter’s moons…

The fact they orbit Jupiter and not Earth challenged the idea of a favored position for the Earth



Observations showed phases for Venus, proving it must orbit the Sun and not the Earth



Showed the Sun had changing sunspots-adding weight against the view of a perfect and unchanging heavens



Summary, 1609: Galileo starts using the telescope for astronomy

Discovers:-phases of Venus (it orbits the Sun)-satellites of Jupiter (they orbit Jupiter)-mountains on the moon-sunspots

Celestial bodies clearly seen to be complex, imperfect and changing-supporting Copernican/heliocentric models

in 1632 Galileo published “Dialogue on Two World Systems” strongly supporting the Copernican system


Galileo Galilei - Relativity

Galileo also added to scientific understanding with some new ideas, realized force is not responsible for motion, but for changes in motion:-

He also postulated - velocity is not absolute, - speed of a falling body is independent of weight, i.e “Galileo's Principle of Equivalence” (later) - a theory of relativity all (mechanical) experiments performed in inertial frames will give the same results

These are cornerstones of Einstein's theories of relativity.


A Few Reminders/ Simple Concepts

Speed conveys how fast one is going

Velocity conveys speed and direction -can change velocity without changing speed

Acceleration/deceleration describes the change in velocity (g=9.8m/s/s)

Momentum describes the product mass x velocity(note mass is amount of matter comprising a body, weight is the sum of accelerating forces on a body)

Force describes a change in momentum


Simple Concepts of Motion

Angular Momentum the product mass x velocity x radius for an object moving in a circle (or orbit)

Jane Turner [4246] PHY 316 (2003 Spring)

Mass always the same, weight depends on force acting on mass


Inertial Frames - Reminder

An inertial frame is one in which under the influence of no forces, an object will remain at rest or in uniform motion (elevator scenario 1)

Accelerating (or rotating) frames not inertial frames.

Given the universal force of gravity, no body can actually be under the influence of no forces. However the concept of inertial frames is still useful since: - effect of gravity can be very small. Thus physical insight can be obtained (e.g. Einstein's Special Theory of Relativity). -valid under Einstein's General Theory of Relativity: a frame in free-fall under the influence of gravity is an inertial frame.


Galileo’s Principle of Equivalence

Inertial and gravitational mass are equivalent

Inertial Massis measure of how strongly the body is accelerated (by A) by a given force.It is the mi in Newton's 2nd-law:Force = mi A

Gravitational Mass measure of how strongly a body is affected by the force of GravityIt is the mg in Newton's universal law of gravitation when the body is a distance R from another body of mass M:Force = G mg M R-2



The inertial mass mi determines how the body accelerates as a results of the application of any force.

The gravitational mass mg determines how the body "feels" a gravitational force (and how much of a gravitational force it generates).

The fact that one can equate the above two forces: mi A = G mg M R-2 Then if mi equals mg then one sees that the acceleration (due to the force of gravity) is independent of mass. Galileo: If all forces apart from gravity can be ignored, all objects fall at the same rate



This equivalence is a mathematical confirmation of Galileo's experiments (cannonball & feather etc) and can be used to derive Kepler's 3rd law of planetary motion

The fact that mi and mg are equal is often referred to as Galileo's Principle of Equivalence

- used and extended by Albert Einstein (1915) as he formulated General Relativity. Indeed the whole of General Relativity rests on Einstein's Principle of Equivalence


Physical Cosmology Arrives

Isaac Newton's discovery that it is gravity which plays the vital role of determining the motion of the planets "signifies the the arrival of the first physical cosmology"

Newton's universal theory of gravity contains the concept of action at a distance

Many scientists did not accept this concept.Newton's law of gravity was not immediately accepted universally.

Instead many scientists preferred to stick with the prevailing idea of the time (e.g. Rene Descartes) that forces work through contact.


Isaac Newton (1643-1727)

Formulated a theory of mechanics & gravity that explained the solar system with remarkable accuracy!

Realized that gravity responsible for the motion of the Moon and planets.

Newton’s law of universal gravitation

Every mass attracts every other massForce drops off with square of distanceKepler’s laws are a direct consequence of Newton’s law of gravity


Orbits under gravity

Gravity also allows us open hyperbolic or parabolic orbits (like those of comets)


Newton Universal Law of Gravitation

Every mass attracts every other mass through the force called gravity

The force of attraction between 2 objects is directly proportional to the product of their masses. Doubling the mass of one body doubles the force of gravity between the two

Fg=GM1M2/r2

r -dist between centersG grav constFg force of grav. attraction


Isaac NewtonIn 1687, Isaac Newton publishes Philosophiae Naturalis Principia Mathematica Stephen Hawking, in A Brief History of Time notes "probably the most important single work ever published in the physical sciences.”

Newtons 3 laws of motion:

1:Every body continues in its state of rest or (straight-line) motion until compelled to do otherwise by an external force (ie. intertial frames) So, if a body is not acted on by any forces, its vel remains const- conservation of momentum

2:Force equals Mass times Acceleration -defines inertial mass as the degree by which a body resistsbeing accelerated by a force

3:To every Force there is an equal (& opposite) Reaction

Remember these!


Isaac Newton

N1:Every body continues in its state of rest or (straight-line) motion until compelled to do otherwise by an external force (inertial frames)Comes from Galilean Relativity- Galileo was the first to state this in his ‘’law of inertia’’

Concept: Frames of references at rest or moving with constant velocity are called inertial frames. Newtons laws hold within these frames of reference

In non-inertial frames you may be fooled into thinking there are forces acting on freely moving bodies


Isaac Newton

N2:Net force is proportional to mass x accelerationAn object subject to n forces experiences a net force

Fnet = Fi = ma (summed over i=1…n) However, a=dv/dt - giving us Fnet = m dv/dt = d (mv)/dt = dp/dt

Vectors, quantities having magnitude and direction, are in boldface


Isaac Newton

N2:Net force is proportional to mass x acceleration

Thus N2 may be expressed as - the net force on an object equals its rate of change of momentum Fnet = dp/dt


Isaac Newton

N3:For every action there is an equal and opposite reaction

action and reaction are forces F12 = -F21


Law of Universal Gravitation

Using his 3 laws of motion+ Keplers 3rd law - Newton was able to find an expression describing the force which holds the planets in their orbits.

Consider a circular orbit of mass m about much larger mass M (M >> m) . Recall K3:-

P2= kr3 Remember this one!

r is dist. between the objects and k is a constant, in circular orbit P is

P = 2 r/v(v is velocity of the mass m)substitute 2nd eqn into first to get

4 2r2/v2 = kr3



4 2r2/v2 = kr3 -rearrange terms and mult. through by m

m v2/r = 4 2m/ kr2

left hand side is recognizable as the centripetal force for circular motion; thus

F = 4 2m/ kr2

must be the grav force keeping m in orbit around M

Now, the force exerted by m on M equal the mag of that exerted by M on m



F = 4 2m/ kr2

Now, the force exerted by m on M equal the mag of that exerted by M on m -thus the form of the eqn should be symmetric w.r.t. exchange of m and M

Expressing the symmetry explicitly and grouping the constants into a new one, G we arrive at the law of universal gravitation found by Newton

F = GMm/r2

G=6.67 x 10-11 m3 kg-1 s-2 SI units



Law of universal gravitation

F = GMm/r2 Remember this one!

Gravitational force follows an inverse square law- doubling separation between two objects, grav attraction drops x 4


Gravity due to Earth

Law of universal gravitation

F = GMm/r2 Remember this one!

Consider mass m falling from height h above Earths surface, earths radius denoted R and mass M

F = G M m/(R + h)2

h << R so F = G M m/R

2

However, F = ma = mg thus g = G M /R

2


Gravity due to Earth

g = G M /R2

M = 5.974 x 1024 kg & R = 6.378x 106 m gives

acceleration due to Earths gravity

g = 9.8 m s-2


Gravitational Potential Energy

Consider now the energy (work) req. to raise mass m to height h against grav. force, ie the change in potential energy of the system

Uf - Ui = U = - F. dr (between ri & rf)

where F is the force vector and ri & rf are initial and final position vectors dr is the infinitesimal change in posn vector

If gravitational force on m is due to mass M then U = G Mm/r2 dr


Gravitational Potential Energy

If gravitational force on m is due to mass M then U = G Mm/r2 dr

evaluate the integral to give Uf - Ui = - G Mm(1/rf - 1/ri) dr

since we are interested in relative change in potential we can choose to define p.e going to zero at infinity , let rf approach infinity so Uf = 0 then (dropping subscripts)

U = - G Mm/r - gravitational p.e. Now, total mechanical energy of a particle is E = 1/2 mv2 - G Mm/r


Escape velocity

E = 1/2 mv2 - G Mm/r

can be used to calculate the escape velocity around mass M (>>m) by equating kinetic energy and grav. force

1/2 mv2 = G Mm/r which may be solved for the velocity

vesc = √(2GM/r) Remember this one!

mass of escaping object does not appear! For earth vesc = 11.2 km/s


Generalization of Keplers work

Newton explained Keplers laws by solving the law of universal gravitation together w/ the laws of motion. Solved a pair of algebraic equations w/ use of calculus

Newtons work showed Keplers first two laws apply to any object orbiting another under the force of gravity, or objects orbiting each other with their center of mass at one focus.


Newtons form of Keplers Third Law

Newton also generalized Kepler’s third law as

P2=42 R3 /G(M1+M2)

Allowed Kepler’s Laws to be applied to moons and (much later) binarystars and extrasolar planets.

Remember this one!


Very good, but not perfect..

Isaac Newton also contributed to science with - invention of the telescopes using mirrors - theories of light (colors & corpuscles) - development of calculus

But Isaac Newton did not get it all right.. e.g. He incorrectly believed space & time were absolute and unaffected by the presence of objects

His character was not perfect either -he had huge academic (& personal) rows with Robert Hooke (re: who 1st discovered the 1/r2 law) Gottfried Wilhelm von Leibnitz (re: who 1st discovered calculus) (e.g. see Steve Hawking “A Brief History of Time”)


Einstein's Principle of Equivalence

Albert Einstein extended Galileo's Principle of Equivalence

(inertial & gravitational masses are equal) i.e. acceleration and gravity cannot be distinguished.

This led to Einstein's Principle of Equivalence The Laws of Physics are the same in a uniformly accelerated reference frame as in a uniform gravitational field

Albert Einstein “The happiest thought of my life”

The whole of General Relativity rests on this (testable) principle


Einstein's Principle of Equivalence

The consequences of the Principle of Equivalence include

- the effects of a gravitational force are indistinguishable from those present in an accelerated reference frame - there is an (accelerating) reference frame in which the effects of gravity are not experienced (falling elevator) - the path of light is "bent" by gravity - clocks run "slow" under the influence of gravity gravitational "redshift" of light waves - gravity affects anything carrying energy (E=mc2)

...more later on GR


Overview

Background reading -Chapter 3

… things to know (for a quiz next week)

-Contributions to astronomy from the ancient Greeks-Ptolemy and his model -Copernicus, how did he change our viewand what was his first model-What is meant by the Copernican & Perfect Cosmological Principles-Kepler, how did he progress from the heliocentric model using circular orbits


Overview

-What are Keplers 3 laws, what factors does the orbital period depend on?

-What were the major achievements of Galileo?

-What was Newtons contribution to astronomy?

-Should be able to remember & use Newtons Law of Universal Gravitation and Keplers Laws(esp. newtons general version of K3)

-& calc escape velocities

jane turner [4246] phy 316 (2003 spring) lecture 3 homework/quiz etc problematic email addresses:...

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